Research Article Modeling the Reflectance of the Lunar...

Research ArticleModeling the Reflectance of the Lunar Regolith bya New Method Combining Monte Carlo Ray Tracing andHapkersquos Model with Application to ChangrsquoE-1 IIM Data

Un-Hong Wong1 Yunzhao Wu2 Hon-Cheng Wong13 Yanyan Liang1 and Zesheng Tang1

1 Space Science Institute Macau University of Science and Technology Macao China2 School of Geographic and Oceanographic Sciences Nanjing University Nanjing 210093 China3 Faculty of Information Technology Macau University of Science and Technology Macao China

Correspondence should be addressed to Hon-Cheng Wong hcwongieeeorg

Received 16 August 2013 Accepted 18 October 2013 Published 12 January 2014

Academic Editors S Falsaperla M Gregoire and K Nemeth

Copyright copy 2014 Un-Hong Wong et alThis is an open access article distributed under theCreativeCommonsAttribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

In this paper wemodel the reflectance of the lunar regolith by a newmethod combiningMonteCarlo ray tracing andHapkersquosmodelThe existing modeling methods exploit either a radiative transfer model or a geometric optical model However the measured datafrom an Interference Imaging spectrometer (IIM) on an orbiter were affected not only by the composition ofminerals but also by theenvironmental factorsThese factors cannot be well addressed by a single model alone Our method implemented Monte Carlo raytracing for simulating the large-scale effects such as the reflection of topography of the lunar soil and Hapkersquos model for calculatingthe reflection intensity of the internal scattering effects of particles of the lunar soil Therefore both the large-scale and microscaleeffects are considered in ourmethod providing amore accurate modeling of the reflectance of the lunar regolith Simulation resultsusing the Lunar Soil Characterization Consortium (LSCC) data and ChangrsquoE-1 elevationmap show that ourmethod is effective anduseful We have also applied our method to ChangrsquoE-1 IIM data for removing the influence of lunar topography to the reflectanceof the lunar soil and to generate more realistic visualizations of the lunar surface

1 Introduction

Determination of the chemical compositions of planetaryregolith is one of the most important objectives for planetaryexploration especially in estimating themineral resources ona solar system body and choosing the landing site for landingprobes on it For this purpose remote sensing has been usedas an efficient approachModeling the reflectance of the lunarregolith is one of key issues in determining the chemicalcompositions of minerals with remotely obtained spectraRadiative transfer models and geometrical optical modelsare commonly used for modeling the reflectance While geo-metrical optical models [1ndash4] are used to investigate therayrsquos reflectionrefraction between particles radiative trans-fer models [5ndash12] focus on the reflectance intensity of dif-ferent compositions of a mineral Monte Carlo methods andray tracing algorithms are widely used in simulating andanalyzing the optically contrast structure of particles with

complex shapes [1ndash4] Lucey et al [13ndash16] proposed a wellapproximation to find out the optical constants from a knownreflectance spectrum using the inverse Hapkes model Pietersand Hiroi [17] tested part of the lunar mineral samples takenback in the Apollo projects The reflectance index of thosetested sample is provided in the Lunar Soil CharacterizationConsortium (LSCC) data

The existing methods for modeling the reflectance of thelunar regolith exploit either a radiative transfer model or ageometric optical model However the measured data froman Interference Imaging spectrometer (IIM) on an orbiterwere affected not only by the composition of minerals butalso by the environmental factors such as the reflection ofthe lunar topography the position of light sources and theposition and orientation of the spectrometer These factorscannot be well addressed by a single model alone Due tothese factors the IIM data cannot be used directly withthe existing models Therefore corrections and adjustments

Hindawi Publishing Corporatione Scientific World JournalVolume 2014 Article ID 457138 14 pageshttpdxdoiorg1011552014457138

2 The Scientific World Journal

are needed [18] Knowledge and experience are requiredto complete these corrections and adjustments but theworkload is very high

In this paper we try to consider these factors bymodelingthe reflectance of the lunar regolith with a new methodcombining Monte Carlo ray tracing for simulating the reflec-tion of topography of the lunar soil and Hapkersquos model forcalculating the reflection intensity of the internal scatteringeffects of particles of the lunar soil Therefore both large-scale and microscale effects are considered in our methodproviding a more accurate modeling of the reflectance of thelunar regolith To the best of the authorrsquos knowledge it isthe first attempt to propose a method considering the abovementioned effects at the same time

Figure 1 gives the schematic view of our method Ourmethod considers the reflectance attributes of the lunar soilthe terrain of the lunar surface the incident light and theposition and viewing angle of the spectrometer and appliesthe existing reflectionmodels to the large-scale scene of lunarsurface Our method simulates the scene when the spectro-meter was measuring the reflectance spectrum and thereflected light of the specific categories of the lunar soil andthe terrain captured by the spectrometer

Our paper is organized as follows A brief review ofHapkersquos model will be given in Section 2 In Section 3 wewill explain some definitions and assumptions of the lunarenvironment and lunar soil including some factors and con-siderations of the design of our method The method and itssimulation steps will be described in Section 4 In Section 5we will provide the simulation results of the Apollo 16 land-ing-site using ChangrsquoE 1 elevation map and the mineralinformation from the LSCC data Two example applicationsof our method will be given in Section 6 Conclusions andfuture work will be provided in Section 7

2 A Brief Review of Hapkersquos RadiativeTransfer Model

Hapkersquos radiative transfer model was proposed by Hapke [6ndash12] The equation we used in our simulation is shown as fol-lows

119903119888=

120596

4 (1205830+ 120583)

119875 (119892) [1 + 119861 (119892)] + 119867 (120583)119867 (1205830) minus 1 (1)

where the dimensionless quantity 119903119888is the radiance coeffi-

cient the variables 120583 and 1205830are the cosines of the reflection

and incidence angles 119892 is the phase angle 119861(119892) is the backscattering function which defines the increase in brightnessof a rough surface with decreasing phase 119875(119892) is the single-particle phase function and the 119867(120583) is the isotropic scat-tering function Details of these functions and variables aredescribed clearly in Lawrence and Luceyrsquos paper [16]

One of the most important parts of (1) is the single-scattering albedo 120596 Optical constants (119899 and 119896) are used inthe calculation of120596 Hapkersquos model actually defines a formulabetween the optical constants and the radiance coefficientsOptical constants also known as the complex indices ofrefraction are used to predict the bidirectional reflectanceof a particulate surface If the optical constants of minerals

in a mixture are known then a reflectance spectrum canbe calculated at arbitrary grain sizes [6] Figure 2 shows thecalculation of the reflectance by Hapkersquos model using opticalconstants as input

However the optical constants of minerals are hard to beobtained Lucey [13] defined the relationship between 119899 and119896 via the Mg-number (Mgmdashthe ratio of Mg to Mg + Fe ona molecular basis is defined as Mg = MgO(MgO + FeO))With Mg 119899 can be represented as a function of 119896mdash119899(119896)Then the unknown numbers in both side of (1) are 119903

119888and 119896

That means 119896 can be calculated when 119903119888is known via inverse

Hapkersquos model Luceyrsquos work provides a way to calculate119896 of a mixture with inverse Hapkersquos model Therefore thereflection spectrum of the LSCC data can be used to obtainthe optical constants (119899 119896) of the samples With the opticalconstants radiance coefficient of any phase angle can becalculated via Hapkersquos model Figure 3 shows the relationshipbetween optical constants and reflectance via Hapkersquos modeland inverse Hapkersquos model

3 Definitions and Assumptions forthe Lunar Soil

Unlike the Earth the density of the atmosphere surroundingthe Moon is tenuous Thus the Moon can be considered avacuum environment It is assumed that there is no reflectionby the atmosphere The view of the lunar landscape isdescribed as ldquothe world of rocksrdquo The only two categoriesof the compositions of the crust found on the lunar surfaceare the highlands Lighter and Anorthositic Surface and thelunar maria Darker and Basaltic Planes Before we explainourmethod themodel representing theMoonrsquos environmentwill be defined and several definitions and assumptions willbe made in the following subsections

31 RefractionReflection of the Lunar Soil Figure 4 showsthe reflected rays of the lunar surfacemdashscattering (diffusion)and the specular reflection The reflection depends on thefollowing

(i) Direction of the light source(ii) Shape of particles(iii) Topology of the surface(iv) Compositions of the soil

Unless there is a large shinny area the contribution ofspecular reflection is negligible in a large-scale landscape [19]Therefore our method focuses on the scattering and ignoresthe specular reflection This assumption will be consideredwhen we simplify (1) in Section 42

When a ray encounters an object the reflectionrefractioncan be analyzed using the Snellrsquos Law Each small bulk of thelunar soil is composed by particles of themineral An incidentray travels between these small particles in the soil Someof the rays (photons) are reflected out of the bulk of lunarsoil while others are refracted inside the mineral and thenfinally absorbed by the mineral (see Figure 5) Monte Carloray tracing of the internal scattering of particleswas presentedin [1ndash4] In the scale of our simulation computational

The Scientific World Journal 3

Mineral samples

Existing models

Incident light

Reflectancespectrum

Spectrometer positionand viewing angle

Light source-incident light

Topography

Reflectance intensityof mineral composition

(Invoked)

Figure 1 The schematic view of our method

Optical constants ReflectanceHapkersquos model

rc(n k)

Figure 2 Hapkersquos model reflectance can be calculated when optical constantssingle scattering albedo are known

Optical constantsHapkersquos model

Inverse Hapkersquos model

Reflectancerc(n(k) k)

Figure 3 After the relationship between 119899 and 119896 are defined optical constants can be found with inverse Hapkersquos model

Light source

Specular (reflection)

Lens

Diffusion (scattering)

Volume of the lunar soil

Image plane

Figure 4 Reflection and scattering of the lunar surface

Figure 5 Complexity of refractionreflection of particles


Regular volume

Irregular volume

Figure 6 Two types of volume structures representing lunar soils (odd and even layers of voxels of the regular volume are shown in differentcolor for a clear view)

complexity of such microscale ray tracing is very high asthere is a huge number of reflections and refractions Weused Hapkersquos radiative transfer model instead to representthe intensity of the reflection of a bulk of the lunar soil Ifthe optical constants are known then the reflectance of anyincident angle and viewing angle (phase angle) can be repro-duced Because we cannot obtain the accurate optical con-stants the optical constants of a sample of the lunar soil areestimated using inverse Hapkersquos model

32 Modeling the Lunar Soil For modeling the lunar soila volume is used to represent the topology of the lunarsurface Each voxel (a unit element of a volume) is assumedto be a small bulk of the lunar soil composed by particlessuch as sands and rocks Two types of volume structures areshown in Figure 6 Regular volume and irregular volume areproposed in our methodThe regular volume is like a regularsampling grid of the lunar soil It is easy to construct easyto be implemented and is efficient for a fast simulation Theirregular volume is more precise to represent the position ofthe actual measured soil samples

Modeling the lunar soil as a volume allows our methodto represent the multilayer reflection model of the lunarsoil For example in a stratified material if a bulk of themineral has different compositionwith respect to themineralat the surface a ray penetratesis refracted through the firstlayer of the lunar surface but then is reflected back while itencounters with an internal bulk of different kind of minerals(as shown in Figure 7) Multilayer reflection is much morecomplex since there will be refraction Therefore we wouldlike to leave this aspect as an extension and future work inanother paper

33 Abstraction of the Model of the Spectrometer In lunarexploration reflected spectra aremeasured by the spectrome-ter installed on an orbiterWe can image that there is a camerainstalled at the bottom of the spacecraft taking pictures whilethe spacecraft flying (orbiting) around theMoon To simplifyit the camera (spectrometer) can be represented by a lens andan image-plane Furthermore since the lens is affected by theviewing angle only (we assume that it will not lose focus)we can use an abstract image-plane and the viewing angleto represent the spectrometer for producing the measuredimages of the reflected spectrum Abstraction of the model

of the spectrometer which captures the reflected spectra ofthe lunar soil is shown in Figure 8

4 Our Method and Simulation Steps

Our method implements Monte Carlo ray tracing andHapkersquos model In this section we will present how we com-binedMonte Carlo ray tracing andHapkersquos model to form theexpression that our implementation is based onThen we willoutline the simulation steps

Monte Carlo ray tracing [20 21] is known as a wellapproximation of the solution of the rendering equation[22] which has been investigated and widely applied toglobal illumination in computer graphics [23ndash25] On theother hand Monte Carlo ray tracing is also widely used insimulating and analyzing the optically contrast structuresof particles with complex shapes Ray tracing describes amethod to produce global illumined images of 3D virtualobjects it traces the path and integrates the emitted energy ofthe light source and reflection between the objects accordingto the principle of optics The rendering equation is anintegral equation formulated on the definition of the bidire-ctional reflectance distribution function (BRDF) plus the self-emittance of surface points at light sources as an initializationfunction Monte Carlo ray tracing solves this integral equa-tion by Monte Carlo integration

BRDF and Monte Carlo ray tracing have been appliedto remote sensing to investigate the characteristics of theelectromagnetic radiation transport between the objects inboth macro-scale and microscale from the surface of theEarth to the internal reflection of particles in a mineral Ourmethod utilizes Monte Carlo ray tracing to trace the path ofthe reflected ray between the light source and the encounteredobject (the lunar surface) and the Hapkersquos radiative transfermodel is used to adjust the intensity of the reflected rayinstead of analyzing the internal scattering of the mineral

41 Definition of the Scene From the point of view of MonteCarlo ray tracing the light source of the scene (the Sun) is ahuge spherical luminaire Each pixel of the measured imagesis an accumulation of all reflected ray hitting the image planeemitted from each small part of the luminaire (see Figure 9)

However even the radius of the Sun is much longer thanthat of the Moon (over 400x) the distance between the Sun


Composition 1Composition 2

Figure 7 Multilayers reflection model

Reflection

Image planeLens

Viewingangle

Reflection

Image planeLens

Abstract image plane

Reflection

Image planeLens

Figure 8 Abstraction of the model of the spectrometer

Image plane

Spherical luminaire

Phase angle

Figure 9 Monte Carlo ray tracing of a spherical luminaire

and the Moon is much longer than the scale of the radiusbetween them (see Figure 10) We can ignore the acreage ofthe measuring area since it is just a very little piece of thesurface of the MoonThe scale of the measuring area is muchsmaller than the radius of the Moon Therefore the mea-suring area can be represented as a dot in this calculationTherange of the phase angle can be calculated by the followingequation

2 arcsin(

RadiusSunDistanceSun-Moon minus RadiusMoon

) lt 001 (2)

As the variation of the phase angle is only 001 degree theincident rays can be treated as parallel rays (see Figure 11) sothat we can revise the definition of the light sourcemdashthe Sunis a parallel light source in the scene rather than a sphericalluminaire In our simulation the range of the phase angle isonly depended on the angle between the horizon of the lunarsurface and the direction to the Sun

42 Reflectance Intensity of the Lunar Surface Reflectionintensity can be determined by the following factors thedirection of the incident light the position of the viewingpoint material and the shape of the object The reflectionequation [22] we used is listed in the following Note that the here is the direction as of two points it is not the single-scattering albedos of the Hapkersquos model

119871119903( 119903) = int

Ω119894

119891119903( 119894997888rarr

119903) 119871119894( 119894) cos 120579

119894d120596119894

(1 2) = (

1997888rarr 2) =

2minus 1

10038161003816100381610038162minus 1

1003816100381610038161003816

(3)

The reflected ray 119871119903is computed by integrating the

incoming ray over a hemisphere centered at a point of thesurface and oriented such that its north pole is aligned withthe surface normal vectorThe incoming radiance along a raydoes not change It obeys the basic laws of geometric optics


Radius of the Sun 696 times 105 km Radius of the Moon 1738km

The Sun-Moon distance 15 times 108 km

Figure 10 The distance between the Sun and the Moon is much longer than the radius of the Sun

Image plane

Figure 11 The Sun as a parallel light source

Lens

Image plane

Light source

Figure 12 The simulation process

assuming that there is no scattering or absorption BRDF119891119903is

a probability distribution function describing the probabilitythat an incoming ray of light is scattered in a randomoutgoingdirection

Then we can calculate the light intensity from the Phongreflection model [26] as follows

119868119901= 119896119886119894119886+ sum

119898isinlights(119896119889(119898sdot ) 119894119898119889

+ 119896119904(119898sdot )

120572

119894119898119904

)

(4)

Equation (4) calculates the illumination of each surface point119868119901 where 119896

119886119894119886is the ambient term the 119896

119889(119898sdot )119894119898119889

is thediffuse term and the 119896

119904(119898sdot)120572

119894119898119904

is the specular term of theincoming light Light is defined as the set of all light sources119898

is the direction vector from the point on the surfacetoward each light source (119898 specifies the light source) isthe normal at this point on the surface

119898is the direction

that a total reflection ray of light would take from this pointon the surface is the direction pointing towards the viewerand 120572 is a shininess constant for this material which is largerfor surfaces that are smoother and more mirror-like Whenthis constant is large the specular highlight is small

Specular reflection is ignored in our simulation Ambientterm accounted for the small amount of light that is scatteredabout the entire scene is also ignored Then only the diffuse

term is remained Instead of using a diffuse reflection con-stant 119896

119889 we use the radiance coefficient 119903

119888of the equation of

Hapkersquosmodel (see (1)) to represent the ratio of themineral indifferent phase anglesThen we derive our equation in whichour method is based on as follows

119868119901= sum

119898isinlights(119903119888(119898sdot ) 119894119898119889

) (5)

43 Simulation Steps The simulation result of (5) containstwo parts the Hapkersquos BDRF contributes the reflectanceintensity of the mineral and the diffuse term of the Phongreflection model contributes the reflectance intensity of thetopography Figure 12 shows the simulation process To findout the contribution of the mineral of the lunar surface apath tracing will be processed starting from each pixel of theimage plane This part is obtained with the following steps

(1) From the position of a pixel of the image plane tracealong a ray to the measured voxel of the volume ofthe lunar soil store the position of themeasured pointand the corresponding mineral

(2) Starting from the encountered point (voxel) traceback to the Sun along the direction of the incident ray

(3) If the incident ray is blocked by other voxel set theresult of the pixel to zero skips the following steps


(4) Else calculate the phase angle using the phase angle tofind out the reflectance spectrumwithHapkersquosmodel

The reflection of the topography is the diffuse term of thePhong reflection model without the 119896

119889mdashonly (

119898sdot ) This

part is calculated with the following steps

(1) Calculate the normalized gradient of the lunar soil(the volume or the elevation map) and use as thenormal vector of the surface

(2) For each measured pointmdashthe encountered voxelfound at the path tracing processmdashdot products thecorresponding incident ray direct and normal vector(both are normalized) to find out the diffusion of thelunar surface

The specular reflection can also be calculated but weignored itThe specular reflection should be considered in thesimulations of a very small scale landscape with a very highresolution image plane or in the case the lunar surface is verysmooth even if there are both mare and highlands exist in themeasured area

5 Simulation of the Surface around Apollo16rsquos Landing-Site

Our method can simulate the reflectance spectrum of severallayers of different mineral composition of the lunar soilHowever we do not have such mineral data Instead weused ChangrsquoE-1 elevation map [27] and the LSCC data todemonstrate how our method can be used to model thereflectance of the lunar regolith ChangrsquoE-1 (CE-1) [28] isa satellite operated in a circular polar orbit about 200 kmabove the lunar surfaceThe LSCCdata (httpwebutkedusimpgidatahtml) includes 9 mare and 10 highland sampleswhich is the only complete ldquoground truthrdquo data of theMoon consisting of both soil reflectance spectra and min-eral abundances The reflectance spectra were measured inRELAB (httpwwwplanetarybrownedurelab) at BrownUniversity

To represent the topography and the mineral of an areaof the lunar surface reflection data of 6 samples of Apollo16 from the LSCC data and CE-1 elevation map were usedThe polar angle and the azimuth angle of the incident lightwere both set to 45 degrees The phase angle between theincident light and the reflected light was calculated and thenthe results were obtained using the reflectance spectrum ofHapkersquos model of the mineral

According to the LSCC data the differences of thereflectance spectrum between the samples of Apollo 16 isbigger than those in other samples For example in the visiblespectrum the biggest difference is 015 at the wavelength of330 nm and is 19 at 780 nm where other samples is about 05at 330 nm and less than 01 at 780 nm By using the samplesof Apollo 16 we can show the effect of the mineral to thereflection clearly

The reflectance spectrum of the samples from Apollo 16project was used and the topography of the lunar soil volumewas reproduced by the laser altimeter of CE-1 of the area

around the Apollo 16 spacecraft landing-site (from 99375 Eto 199375 E longitude 50625 S to 150625 S latitude) Theresolution of the elevation map [27] is 256 times 256

The diffusion of the topography was calculated usingthe normal vectors of the surface Normal vectors can becalculated using the elevation map It is a common way toinitialize a unit vector pointing along the 119911 axis (we assumedthat the surface is on the 119909-119910 plane and 119911 direction points tothe upward) define its length (the unit length) with the samescale as the scale of the 119909-119910 resolution of the elevation mapThen the surface normal vector can be found out by the unitvector minus the gradient of 119909 dimension and 119910 dimensionof the elevation map The setup of the topography of CE-1elevation map are shown in Figure 13 The simulation resultof the terrain with diffusion is shown in Figure 14 Note thatthe normal vectors must be normalized before using in thecalculation of the diffusion

In the following subsections simulation results withdifferent distributions of the mineral will be shown Testingparameters are set as follows the spectrometer was set atthe height of 10000 km height facing to the center of thearea and the image plane is parallel to the lunar surface Theincident angle and azimuth angle of the Sun were both set to45 degreesThe reflectance of the wavelength 120582 = 645 nmwastested in the simulation

51 Uniform Distribution According to the LSCC data thereare 4 kinds of particle size of each powdered sample Evenfor the same mineral composition samples with differentparticle size provide different spectra Therefore we actuallygot 6 times 4 = 24 minerals (different spectra) of the Apollo16 samples This test case assumed that those 24 kinds ofthe mineral distributed in the area uniformly (with sameprobability) everywhere The simulation result of differentsampling rates is shown in Figure 15 From this figure we canrealize that the result of the terrain with diffusion only issmooth and the result with the reflectance of the differenceof the samples shows the ldquotexturerdquo of the lunar soil

52 Location Based Distribution In this simulation 6 kindsof the spectrum of the samples were placed in 6 locations inthe area During sampling each point of the surface producedreflection intensity with the probability of which kind of themineral it is The probability is based on the distance ofbetween that point and the six locations of the samples Justlike when we found a mineral in one specific location thenwe considered that the near-by regionmight contain a similarmineral as well After the mineral had been determined theparticle size of that kind ofmineral was chosen randomlyThesimulation result is shown in Figure 16

53 Analysis of the Simulation Results Figures 15 and 16show the images of a scene of an orbiter that measuring theApollo 16 landing-site were reproduced by our simulationsThe simulation results were affected by the following fac-torsparameters

(i) The position of the luminairemdashincident light angle(ii) The position of the orbitermdashreflection angle


x

y

z

Figure 13 The setup of the topography of CE-1 elevation map

50 100 150 200 250

50

100

150

200

250

04

045

05

055

06

065

07

075

08

085

09

Figure 14 Simulation result of the terrain with diffusion (size 256 times 256)

(iii) Themineral of the volumemdashreflectance spectrumcal-culated from Hapkersquos model

(iv) The topology of the volumemdashdiffusion and the block-ed rays

From Figures 15 and 16 we can realize that if the diffusionand the reflectance from the radiative transfer model werenot considered we could only produce a smooth surface Ifonly the reflectance from the radiative transfer model wereconsidered we could only produce an image with the darkerand lighter regions Such image was not able to show anytexture (topography details) of the surface Once both the dif-fusion and the reflectance from the radiative transfer modelare combined the reflectance of the rocky lunar surface isrevealed The simulation results with single wavelength (120582 =

645 nm) are shown in this paper but our method is able togenerate the results with multiple wavelengths

Due to the large scale of the width length and height ofthe scene and the position of the Sun there is no shadowshown in the simulation results (Figures 15 and 16) Figure17 demonstrates a scene which shadows are consideredwhen the incident light comes along a big incident angleBlocked incident lights will produce a zero reflectance to thespectrometer Shadows and blocked incident lights are notonly for generating a realistic reflectance image but also in

analyzing the measured data as we can determine which darkparts were caused by low reflectance rate of the lunar soil orby block incident lights

In all the simulation results shown in Figures 15 and 16the phase angle was varying from 329 degrees to 570 degreesAccording to the results of the Hapkersquos model the reflectanceshould be varying with different phase angles However itdose not show the differences very clearly in these resultsIn our analysis we found that the reflectance were actuallyvarying but with the order of 10

minus3 (see Figure 18 for thevarying of the reflectance index of a simulation result with theldquo61141rdquo sample)Thus the varying of the reflectance is too lessto be noticed in the resulting images in Figures 15 and 16

6 Applications

In this section we will show two example applications of ourmethodThe first one is on how to use our method to removethe reflection of the topography in ChangrsquoE-1 IIM data Thereflection will confuse the further analysis of the data Thesecond one is on how to generate more clear visualizations ofthe topography of the lunar soil

61 Removal of the Reflection of the Topography in ChangrsquoE-1IIMData Here we apply ourmethod to ChangrsquoE-1 IIM data


50 100 150 200 250

50

100

150

200

250

04

045

05

055

06

065

07

075

08

085

09

(a)

50 100 150 200 250

50

100

150

200

250

01

012

014

016

018

02

022

(b)

Figure 15 Simulation results of the uniform distribution of the samples (size 256 times 256) (a) diffusion only (b) diffusion and reflectance ofthe samples (sample rate = 2000)

(httpmoonbaoaccn) IIM data measured by the spec-trometer contain the impacts from the mineral compositionand the shape of the topography In developing our methodboth the reflectance of the topography and the mineral wereconsidered Therefore an inverse process of the simulationcan be used to remove the reflection of the topography fromthe raw measured spectra According to (5) of our methodwe can obtain the approximation of the spectrum excludingthe effect from the topography by the following formulas

sum

119898isinlights119903119888119898

= sum

119898isinlights

119868119901119898

(119898sdot ) 119894119898119889

119903119888119898

=

119868119901119898

119908119898(119898sdot ) 119894119898119889

(6)

where119908119898is the weighting factorfunction to adjust the effect

rate of the topography As an example let us take a look to

a lunar surface spectrum measured by CE-1 Figure 19 showsa track of the lunar surface spectrum (120582 = 705 nm) withinthe longitude 158018 E to 164707 E and latitude 50619 Sto 150495 S the diffusion of our simulation and the resultof the spectrum divided by the diffusion The diffusion ofeach point was calculated with the same incident angle andazimuth angle as the timewhile the orbiter wasmeasuring thespectrum Without considering the noise and other reasonsthat may course the variation of the measured we onlydivided the measured data with the diffusion (119908

119898= 1)

Then we obtained the results shown in Figure 19 It can benoticed that some slopes and peaks of themeasured data wereadjusted to horizontal lines It is probable that minerals atthose regions have similar reflectance as their neighboringminerals And they are possible the same kind of mineralsHowever the intensity of the reflected light is different sincethey were lying on inclines Therefore an inverse process ofour method can be useful in removing the influence of lunar


50 100 150 200 250

50

100

150

200

250

018

02

022

024

026

028

03

032

034

(a)

50 100 150 200 250

50

100

150

200

250

01

012

014

016

018

02

022

024

026

(b)

Figure 16 Simulation results of the location based distribution of the minerals (sample rate = 1000 size 256 times 256) (a) the reflectance of thedistribution of the samples (b) final result of the terrain with diffusion

Light source


(a)

50 100 150 200 250

50

100

150

200

250

0

005

01

015

02

025

03

035

04

(b)

Figure 17 Simulation result with shadows caused by the blocked ray (size 256 times 256)

topography to the reflectance of the lunar soil The 2D results(119909-119910 profile) of the same region is shown in Figure 20

62 Visualization of the Lunar Surface Visualization is apowerful tool to understand and analyze the characteristicsof volume data For example 3D imaging of CT (ComputedTomography) data using volume visualization techniquesnowadays plays an important role in daily use in hospitalsIn this example application we show how better visualizationresults of the lunar surface can be obtained using the resultssimulated by our method In our method the lunar soilis modeled as a volume Volumetric ray casting [29] canbe applied to the lunar soil volume to generate a 3D viewof the lunar surface without constructing meshes Besides

special visualization effects such as showing the differentcomposition influences of the lunar surface can be provided

Volumetric ray casting [29] is one of the useful visual-ization algorithms for visualizing volume data revealing theinternal structures of the data Figure 21 shows the concept ofvolumetric ray casting shooting rays from the viewing pointto the volume data then sampling the voxels along each rayand compositing the value and with shading to generate theimages For different purpose and visual effects the calcula-tion of composition and shading can be different A commonway is to apply a user-defined transfer function to map thevalues of the data to specific RGBA color to reveal the inter-esting parts of the data For further information about volu-metric ray casting and transfer functions please refer to [29]


32 37 42 47 52 570219

02195

022

02205

0221

02215

0222

(a)

32 37 42 47 52 5701605

0161

01615

0162

(b)

32 37 42 47 52 570209

02095

021

02105

0211

02115

(c)

32 37 42 47 52 5701745

0175

01755

0176

01765

(d)

Figure 18The phase angle from 329 degrees to 570 degrees of the spectrum of the ldquo61141rdquo sample (119909-axis is reduced reflectances and 119910-axisis phase angle (degrees))

200 300 400 500 600 700 800075085095

Diffusion

(a)

200 300 400 500 600 700 800005

015

025

Spectrum of 705

(b)

200 300 400 500 600 700 800005015025

Spectrumdiffusion

(c)

Figure 19 The 1D results of removing the influence of the topography applying our model (in all figures 119909-axis is wavelength (nm) and119910-axix is the reflectance) (a) simulated diffusion results (b) 119910-profile of CE-1 IIM data (120582 = 705 nm) (c) the result using the inverse modelof our method to CE-1 IIM data The featuring parts to show the changes after dividing the diffusion are circled


20 40 60 80 100

200

400

600

800

1000

1200

1400

005

01

015

02

025

03

035

(a)

200

400

600

800

1000

1200

1400

20 40 60 80 100

065

07

075

08

085

09

095

(b)

200

400

600

800

1000

1200

1400

20 40 60 80 100

01

015

02

025

03

035

04

(c)

Figure 20 The 2D results of removing the influence of the topography applying our model (a) 119909-119910 profile of CE-1 IIM data (120582 = 705 nm)(b) diffusion intensity calculated by CE-1 elevation map of the same region (c) result obtained by using the inverse process of our method toCE-1 IIM data

VoxelSample points

Ray

Viewing point

Figure 21 The concept of volumetric ray casting

Figures 22 and 23 show some visualization results of thelunar soil model In Figure 23 we simulated if there wereinternal (multilayer) compositions of the soil Since we didnot have such multilayer information the composition of theunder layer was constructed by a random number generatingfunction The visualization results shown in Figure 22 weregenerated by simply adjusting the normal vectors to makethe contour of the topography being shown clear while theresults of Figure 23 were generated by mapping with 1D and2D transfer functions as well as by applying different shadingmethods to enhance the varying features of topography of thelunar surface

7 Conclusion and Future Work

In this paper we model the reflectance of the lunar regolithby a new method combining Monte Carlo ray tracing and

Hapkersquos model Both large-scale effects such as the reflectionof topography of the lunar soil and microscale effects suchas the reflection intensity of the internal scattering effects ofparticles of the lunar soil are considered in our method Tothe best of the authorrsquos knowledge it is the first attempt topropose a method considering the above effects at the sametime giving a more accurate modeling of the reflectance ofthe lunar regolith compared to those existing methods usingeither a radiative transfer model or a geometrical opticalmodel Our method aims to provide an attempt to apply thecalculated optical constants of samples to the scene of thelunar environment including the factors of the influence ofthe terrain as well as the light source As a result the incidentlight terrain categories of the lunar soil viewing position areall included in our method Even shadows can be reproducedvia the ray tracing algorithm Phong reflection model wasused to calculate the reflection intensity of the topography


(a) (b)

Figure 22 Visualization results of the lunar soil volume Topography details are enhanced using volumetric ray casting The result on theright is with shadows and the reflectance of the location based distribution of the minerals

(a) (b)

Figure 23 Visualization results of the lunar soil volume Mineral beneath the surface is generated using a random function (a) the resultwithout shading effect (b) the result with shading effect

The concept of our method follows Monte Carlo ray tracingto find out the path and the phase angle of the reflectedrays Thus instead of a constant factor Hapkersquos radiativetransfermodel was used to represent the reflection ratio of themineral Simulation results of the lunar surface around theApollo 16 landing-site are shown to demonstrate ourmethodReflectance spectrumof theApollo 16 samples from the LSCCdata and the topography data from CE-1 elevation map wereused We also apply our method to ChangrsquoE-1 IIM data forremoving the influence of lunar topography to the reflectanceof the lunar soil and to generate more clear visualizations ofthe lunar surface

We model the lunar soil as volume then the internalstructures or multi-layers structures of the lunar surface can

be represented Therefore our method can be extended tohandle if there is a ray refracted into the soil and then reflectedback while the ray encountered a different mineral beneaththe lunar surface In this practice only CE-1 elevation mapwas used in modeling the lunar soil The model of the lunarsoil can be adjusted and improved if more measured dataof the lunar surfacesoil are invoked such as the slope mapand roughness map from the Lunar Reconnaissance Orbiter(LRO) of the lunar surface We also look forward to applyingour method using the data from other lunar explorationprojects such as LRO data and KAGUYA data

Hapkersquos radiative transfer was used as BDRF in ourmethod to reproduce the reflection index of themineral Cal-culation can only provide approximation results which may


contain errors Therefore the method can be refined in thefuture or use another BDRF to obtainmore precise simulationresults

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The work was supported by the Science and TechnologyDevelopment Fund of Macao SAR (0182010A 0042011A10482012A2 and 0802012A3) the Open Research Fund ofKey Laboratory of Digital Earth Center for Earth Obser-vation and Digital Earth Chinese Academy of Sciences(2011LDE006) and the Open Research Funding Program ofKLGIS (2011A09) The authors would also like to thank theteams of USGSmineral library and RELAB at Brown Univer-sity LSCC data and ChangrsquoE-1 data for providing these dataSpecial thanks to anonymous reviewers for their constructiveand valuable comments that helped us to improve the paper

References

[1] W M Grundy S Doute and B Schmitt ldquoA Monte Carlo ray-tracing model for scattering and polarization by large particleswith complex shapesrdquo Journal of Geophysical Research E vol105 no 12 pp 29291ndash29314 2000

[2] Y Grynko and Y Shkuratov ldquoRay tracing sumulation of lightscattering by spherical clusters consisting of particles with dif-ferent shapesrdquo Journal of Quantitative Spectroscopy amp RadiativeTransfer vol 106 no 1ndash3 pp 56ndash62 2007

[3] M Mikrenska P Koulev J-B Renard E Hadamcik and J-C Worms ldquoDirect simulation Monte Carlo ray tracing modelof light scattering by a class of real particles and comparisonwith PROGRA2 experimental resultsrdquo Journal of QuantitativeSpectroscopy amp Radiative Transfer vol 100 no 1ndash3 pp 256ndash2672006

[4] D Stankevich andY Shkuratov ldquoMonteCarlo ray-tracing simu-lation of light scattering in particulatemedia with optically con-trast structurerdquo Journal of Quantitative SpectroscopyampRadiativeTransfer vol 87 no 3-4 pp 289ndash296 2004

[5] K Lumme and E Bowell ldquoRadiative transfer in the surfaces ofatmosphereless bodies 1TheoryrdquoAstronomical Journal vol 86pp 1694ndash1704 1981

[6] B Hapke ldquoBidirectional reflectance spectroscopy 1 TheoryrdquoJournal of Geophysical Research vol 86 pp 3039ndash3054 1981

[7] B Hapke ldquoBidirectional reflectance spectroscopy 3 Correctionformacroscopic roughnessrdquo Icarus vol 59 no 1 pp 41ndash59 1984

[8] BHapke ldquoBidirectional reflectance spectroscopy 4The extinc-tion coefficient and the opposition effectrdquo Icarus vol 67 no 2pp 264ndash280 1986

[9] B Hapke Theory of Reectance and Emittance SpectroscopyCambridge University Press Cambridge Mass USA 1993

[10] B Hapke ldquoSpace weathering fromMercury to the asteroid beltrdquoJournal of Geophysical Research E vol 106 no 5 pp 10039ndash10073 2001

[11] B Hapke ldquoBidirectional reflectance spectroscopy 5The coher-ent backscatter opposition effect and anisotropic scatteringrdquoIcarus vol 157 no 2 pp 523ndash534 2002

[12] B Hapke Theory of Reflectance and Emittance SpectroscopyCambridge University Press Cambridge Mass USA 2nd edi-tion 2012

[13] P G Lucey ldquoModel near-infrared optical constants of olivineand pyroxene as a function of iron contentrdquo Journal of Geophys-ical Research E vol 103 no 1 pp 1703ndash1713 1998

[14] P G Lucey ldquoMineral maps of the Moonrdquo Geophysical ResearchLetters vol 31 no 8 Article ID L08701 2004

[15] P G Lucey ldquoRadiative transfer modeling of the effect of mine-ralogy on some empirical methods for estimating iron concen-tration from multispectral imaging of the moonrdquo Journal ofGeophysical Research E vol 111 no 8 Article ID E08003 2006

[16] S J Lawrence and P G Lucey ldquoRadiative transfer mixing mod-els of meteoritic assemblagesrdquo Journal of Geophysical ResearchE vol 112 no 7 Article ID E07005 2007

[17] C M Pieters and T Hiroi ldquoRELAB (Reflectance ExperimentLaboratory) a NASA multiuser spectroscopy facilityrdquo in Pro-ceedings of the 35th Lunar and Planetary Science Conference2004 abstract no 1720

[18] Y Wu B Xue B Zhao et al ldquoGlobal estimates of lunar ironand titanium contents from the Changrsquo E-1 IIM datardquo Journal ofGeophysical Research E vol 117 no 2 Article ID E02001 2012

[19] H E Bennett ldquoSpecular reflectance of aluminized ground glassand the height distribution of surface irregularitiesrdquo Journal ofthe Optical Society of America vol 53 pp 1389ndash1394 1963

[20] J Spanier and E M Gelbard Monte Carlo Principles andNeutron Transport Problems Addison-Wesley Reading MassUSA 1969

[21] L Carter andECashwellParticle-Transport Simulationwith theMonte Carlo Methods US Department of Energy 1975

[22] J T Kajiya ldquoThe rendering equationrdquo Computer Graphics vol20 no 4 pp 143ndash150 1986

[23] P ShirleyRealistic Ray Tracing A K Peters NatickMass USA2000

[24] H W Jensen J Arvo P Dutre et al ldquoMonte Carlo ray tracingrdquoin Proceedings of the ACM SIGGRAPH Conference on ComputerGraphics and Interactive Techniques Course Notes 44 2003httpgeometrycaltechedusimkeenanmcrt-sg03cpdf

[25] P Dutre P Bekaert and K Bala Advanced Global IlluminationA K Peters Natick Mass USA 2003

[26] B T Phong ldquoIllumination for computer generated picturesrdquoCommunications of the ACM vol 18 no 6 pp 311ndash317 1975

[27] Z Cai C Zheng Z Tang and D Qi ldquoLunar digital elevationmodel and elevation distribution model based on ChangrsquoE-1LAM datardquo Science China Technological Sciences vol 53 no 9pp 2558ndash2568 2010

[28] Y Zheng Z Ouyang C Li J Liu and Y Zou ldquoChinarsquos lunarexploration program present and futurerdquo Planetary and SpaceScience vol 56 no 7 pp 881ndash886 2008

[29] M Hadwiger J Kniss C Rezk-Salama D Weiskopf and KEngel Real Time Volume Graphics A K Peters Natick MassUSA 2006

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ClimatologyJournal of

EcologyInternational Journal of


EarthquakesJournal of


Hindawi Publishing Corporationhttpwwwhindawicom

Applied ampEnvironmentalSoil Science

Volume 2014

Mining


Journal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

International Journal of

Geophysics

OceanographyInternational Journal of


Journal of Computational Environmental SciencesHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofPetroleum Engineering


GeochemistryHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Atmospheric SciencesInternational Journal of


OceanographyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in


MineralogyInternational Journal of


MeteorologyAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Paleontology JournalHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014


Geological ResearchJournal of


Geology Advances in


are needed [18] Knowledge and experience are requiredto complete these corrections and adjustments but theworkload is very high

In this paper we try to consider these factors bymodelingthe reflectance of the lunar regolith with a new methodcombining Monte Carlo ray tracing for simulating the reflec-tion of topography of the lunar soil and Hapkersquos model forcalculating the reflection intensity of the internal scatteringeffects of particles of the lunar soil Therefore both large-scale and microscale effects are considered in our methodproviding a more accurate modeling of the reflectance of thelunar regolith To the best of the authorrsquos knowledge it isthe first attempt to propose a method considering the abovementioned effects at the same time

Figure 1 gives the schematic view of our method Ourmethod considers the reflectance attributes of the lunar soilthe terrain of the lunar surface the incident light and theposition and viewing angle of the spectrometer and appliesthe existing reflectionmodels to the large-scale scene of lunarsurface Our method simulates the scene when the spectro-meter was measuring the reflectance spectrum and thereflected light of the specific categories of the lunar soil andthe terrain captured by the spectrometer

Our paper is organized as follows A brief review ofHapkersquos model will be given in Section 2 In Section 3 wewill explain some definitions and assumptions of the lunarenvironment and lunar soil including some factors and con-siderations of the design of our method The method and itssimulation steps will be described in Section 4 In Section 5we will provide the simulation results of the Apollo 16 land-ing-site using ChangrsquoE 1 elevation map and the mineralinformation from the LSCC data Two example applicationsof our method will be given in Section 6 Conclusions andfuture work will be provided in Section 7

2 A Brief Review of Hapkersquos RadiativeTransfer Model

Hapkersquos radiative transfer model was proposed by Hapke [6ndash12] The equation we used in our simulation is shown as fol-lows

119903119888=

120596

4 (1205830+ 120583)

119875 (119892) [1 + 119861 (119892)] + 119867 (120583)119867 (1205830) minus 1 (1)

where the dimensionless quantity 119903119888is the radiance coeffi-

cient the variables 120583 and 1205830are the cosines of the reflection

and incidence angles 119892 is the phase angle 119861(119892) is the backscattering function which defines the increase in brightnessof a rough surface with decreasing phase 119875(119892) is the single-particle phase function and the 119867(120583) is the isotropic scat-tering function Details of these functions and variables aredescribed clearly in Lawrence and Luceyrsquos paper [16]

One of the most important parts of (1) is the single-scattering albedo 120596 Optical constants (119899 and 119896) are used inthe calculation of120596 Hapkersquos model actually defines a formulabetween the optical constants and the radiance coefficientsOptical constants also known as the complex indices ofrefraction are used to predict the bidirectional reflectanceof a particulate surface If the optical constants of minerals

in a mixture are known then a reflectance spectrum canbe calculated at arbitrary grain sizes [6] Figure 2 shows thecalculation of the reflectance by Hapkersquos model using opticalconstants as input

However the optical constants of minerals are hard to beobtained Lucey [13] defined the relationship between 119899 and119896 via the Mg-number (Mgmdashthe ratio of Mg to Mg + Fe ona molecular basis is defined as Mg = MgO(MgO + FeO))With Mg 119899 can be represented as a function of 119896mdash119899(119896)Then the unknown numbers in both side of (1) are 119903

119888and 119896

That means 119896 can be calculated when 119903119888is known via inverse

Hapkersquos model Luceyrsquos work provides a way to calculate119896 of a mixture with inverse Hapkersquos model Therefore thereflection spectrum of the LSCC data can be used to obtainthe optical constants (119899 119896) of the samples With the opticalconstants radiance coefficient of any phase angle can becalculated via Hapkersquos model Figure 3 shows the relationshipbetween optical constants and reflectance via Hapkersquos modeland inverse Hapkersquos model

3 Definitions and Assumptions forthe Lunar Soil

Unlike the Earth the density of the atmosphere surroundingthe Moon is tenuous Thus the Moon can be considered avacuum environment It is assumed that there is no reflectionby the atmosphere The view of the lunar landscape isdescribed as ldquothe world of rocksrdquo The only two categoriesof the compositions of the crust found on the lunar surfaceare the highlands Lighter and Anorthositic Surface and thelunar maria Darker and Basaltic Planes Before we explainourmethod themodel representing theMoonrsquos environmentwill be defined and several definitions and assumptions willbe made in the following subsections

31 RefractionReflection of the Lunar Soil Figure 4 showsthe reflected rays of the lunar surfacemdashscattering (diffusion)and the specular reflection The reflection depends on thefollowing

(i) Direction of the light source(ii) Shape of particles(iii) Topology of the surface(iv) Compositions of the soil

Unless there is a large shinny area the contribution ofspecular reflection is negligible in a large-scale landscape [19]Therefore our method focuses on the scattering and ignoresthe specular reflection This assumption will be consideredwhen we simplify (1) in Section 42

When a ray encounters an object the reflectionrefractioncan be analyzed using the Snellrsquos Law Each small bulk of thelunar soil is composed by particles of themineral An incidentray travels between these small particles in the soil Someof the rays (photons) are reflected out of the bulk of lunarsoil while others are refracted inside the mineral and thenfinally absorbed by the mineral (see Figure 5) Monte Carloray tracing of the internal scattering of particleswas presentedin [1ndash4] In the scale of our simulation computational


Mineral samples

Existing models

Incident light

Reflectancespectrum



Topography


(Invoked)



rc(n k)






Light source


Lens



Image plane




Regular volume

Irregular volume
















Reflection

Image planeLens

Viewingangle

Reflection

Image planeLens


Reflection

Image planeLens


Image plane

Spherical luminaire

Phase angle



2 arcsin(


) lt 001 (2)



119871119903( 119903) = int

Ω119894

119891119903( 119894997888rarr

119903) 119871119894( 119894) cos 120579

119894d120596119894

(1 2) = (

1997888rarr 2) =

2minus 1

10038161003816100381610038162minus 1

1003816100381610038161003816

(3)







Image plane


Lens

Image plane

Light source





119868119901= 119896119886119894119886+ sum

119898isinlights(119896119889(119898sdot ) 119894119898119889

+ 119896119904(119898sdot )

120572

119894119898119904

)

(4)



119889(119898sdot )119894119898119889


119904(119898sdot)120572

119894119898119904










119868119901= sum

119898isinlights(119903119888(119898sdot ) 119894119898119889

) (5)








119889mdashonly (

119898sdot ) This


















x

y

z


50 100 150 200 250

50

100

150

200

250

04

045

05

055

06

065

07

075

08

085

09










6 Applications




50 100 150 200 250

50

100

150

200

250

04

045

05

055

06

065

07

075

08

085

09

(a)

50 100 150 200 250

50

100

150

200

250

01

012

014

016

018

02

022

(b)



sum

119898isinlights119903119888119898

= sum

119898isinlights

119868119901119898

(119898sdot ) 119894119898119889

119903119888119898

=

119868119901119898

119908119898(119898sdot ) 119894119898119889

(6)




119898= 1)



50 100 150 200 250

50

100

150

200

250

018

02

022

024

026

028

03

032

034

(a)

50 100 150 200 250

50

100

150

200

250

01

012

014

016

018

02

022

024

026

(b)


Light source


(a)

50 100 150 200 250

50

100

150

200

250

0

005

01

015

02

025

03

035

04

(b)







32 37 42 47 52 570219

02195

022

02205

0221

02215

0222

(a)

32 37 42 47 52 5701605

0161

01615

0162

(b)

32 37 42 47 52 570209

02095

021

02105

0211

02115

(c)

32 37 42 47 52 5701745

0175

01755

0176

01765

(d)


200 300 400 500 600 700 800075085095

Diffusion

(a)

200 300 400 500 600 700 800005

015

025

Spectrum of 705

(b)

200 300 400 500 600 700 800005015025

Spectrumdiffusion

(c)



20 40 60 80 100

200

400

600

800

1000

1200

1400

005

01

015

02

025

03

035

(a)

200

400

600

800

1000

1200

1400

20 40 60 80 100

065

07

075

08

085

09

095

(b)

200

400

600

800

1000

1200

1400

20 40 60 80 100

01

015

02

025

03

035

04

(c)


VoxelSample points

Ray

Viewing point







(a) (b)


(a) (b)










Acknowledgments


References







































Volume 2014

Mining


Journal of



Geophysics







Journal of




Advances in











Geology Advances in


Mineral samples

Existing models

Incident light

Reflectancespectrum



Topography


(Invoked)



rc(n k)






Light source


Lens



Image plane




Regular volume

Irregular volume
















Reflection

Image planeLens

Viewingangle

Reflection

Image planeLens


Reflection

Image planeLens


Image plane

Spherical luminaire

Phase angle



2 arcsin(


) lt 001 (2)



119871119903( 119903) = int

Ω119894

119891119903( 119894997888rarr

119903) 119871119894( 119894) cos 120579

119894d120596119894

(1 2) = (

1997888rarr 2) =

2minus 1

10038161003816100381610038162minus 1

1003816100381610038161003816

(3)







Image plane


Lens

Image plane

Light source





119868119901= 119896119886119894119886+ sum

119898isinlights(119896119889(119898sdot ) 119894119898119889

+ 119896119904(119898sdot )

120572

119894119898119904

)

(4)



119889(119898sdot )119894119898119889


119904(119898sdot)120572

119894119898119904










119868119901= sum

119898isinlights(119903119888(119898sdot ) 119894119898119889

) (5)








119889mdashonly (

119898sdot ) This


















x

y

z


50 100 150 200 250

50

100

150

200

250

04

045

05

055

06

065

07

075

08

085

09










6 Applications




50 100 150 200 250

50

100

150

200

250

04

045

05

055

06

065

07

075

08

085

09

(a)

50 100 150 200 250

50

100

150

200

250

01

012

014

016

018

02

022

(b)



sum

119898isinlights119903119888119898

= sum

119898isinlights

119868119901119898

(119898sdot ) 119894119898119889

119903119888119898

=

119868119901119898

119908119898(119898sdot ) 119894119898119889

(6)




119898= 1)



50 100 150 200 250

50

100

150

200

250

018

02

022

024

026

028

03

032

034

(a)

50 100 150 200 250

50

100

150

200

250

01

012

014

016

018

02

022

024

026

(b)


Light source


(a)

50 100 150 200 250

50

100

150

200

250

0

005

01

015

02

025

03

035

04

(b)







32 37 42 47 52 570219

02195

022

02205

0221

02215

0222

(a)

32 37 42 47 52 5701605

0161

01615

0162

(b)

32 37 42 47 52 570209

02095

021

02105

0211

02115

(c)

32 37 42 47 52 5701745

0175

01755

0176

01765

(d)


200 300 400 500 600 700 800075085095

Diffusion

(a)

200 300 400 500 600 700 800005

015

025

Spectrum of 705

(b)

200 300 400 500 600 700 800005015025

Spectrumdiffusion

(c)



20 40 60 80 100

200

400

600

800

1000

1200

1400

005

01

015

02

025

03

035

(a)

200

400

600

800

1000

1200

1400

20 40 60 80 100

065

07

075

08

085

09

095

(b)

200

400

600

800

1000

1200

1400

20 40 60 80 100

01

015

02

025

03

035

04

(c)


VoxelSample points

Ray

Viewing point







(a) (b)


(a) (b)










Acknowledgments


References







































Volume 2014

Mining


Journal of



Geophysics







Journal of




Advances in











Geology Advances in


Regular volume

Irregular volume
















Reflection

Image planeLens

Viewingangle

Reflection

Image planeLens


Reflection

Image planeLens


Image plane

Spherical luminaire

Phase angle



2 arcsin(


) lt 001 (2)



119871119903( 119903) = int

Ω119894

119891119903( 119894997888rarr

119903) 119871119894( 119894) cos 120579

119894d120596119894

(1 2) = (

1997888rarr 2) =

2minus 1

10038161003816100381610038162minus 1

1003816100381610038161003816

(3)







Image plane


Lens

Image plane

Light source





119868119901= 119896119886119894119886+ sum

119898isinlights(119896119889(119898sdot ) 119894119898119889

+ 119896119904(119898sdot )

120572

119894119898119904

)

(4)



119889(119898sdot )119894119898119889


119904(119898sdot)120572

119894119898119904










119868119901= sum

119898isinlights(119903119888(119898sdot ) 119894119898119889

) (5)








119889mdashonly (

119898sdot ) This


















x

y

z


50 100 150 200 250

50

100

150

200

250

04

045

05

055

06

065

07

075

08

085

09










6 Applications




50 100 150 200 250

50

100

150

200

250

04

045

05

055

06

065

07

075

08

085

09

(a)

50 100 150 200 250

50

100

150

200

250

01

012

014

016

018

02

022

(b)



sum

119898isinlights119903119888119898

= sum

119898isinlights

119868119901119898

(119898sdot ) 119894119898119889

119903119888119898

=

119868119901119898

119908119898(119898sdot ) 119894119898119889

(6)




119898= 1)



50 100 150 200 250

50

100

150

200

250

018

02

022

024

026

028

03

032

034

(a)

50 100 150 200 250

50

100

150

200

250

01

012

014

016

018

02

022

024

026

(b)


Light source


(a)

50 100 150 200 250

50

100

150

200

250

0

005

01

015

02

025

03

035

04

(b)







32 37 42 47 52 570219

02195

022

02205

0221

02215

0222

(a)

32 37 42 47 52 5701605

0161

01615

0162

(b)

32 37 42 47 52 570209

02095

021

02105

0211

02115

(c)

32 37 42 47 52 5701745

0175

01755

0176

01765

(d)


200 300 400 500 600 700 800075085095

Diffusion

(a)

200 300 400 500 600 700 800005

015

025

Spectrum of 705

(b)

200 300 400 500 600 700 800005015025

Spectrumdiffusion

(c)



20 40 60 80 100

200

400

600

800

1000

1200

1400

005

01

015

02

025

03

035

(a)

200

400

600

800

1000

1200

1400

20 40 60 80 100

065

07

075

08

085

09

095

(b)

200

400

600

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1000

1200

1400

20 40 60 80 100

01

015

02

025

03

035

04

(c)


VoxelSample points

Ray

Viewing point







(a) (b)


(a) (b)










Acknowledgments


References







































Volume 2014

Mining


Journal of



Geophysics







Journal of




Advances in











Geology Advances in




Reflection

Image planeLens

Viewingangle

Reflection

Image planeLens


Reflection

Image planeLens


Image plane

Spherical luminaire

Phase angle



2 arcsin(


) lt 001 (2)



119871119903( 119903) = int

Ω119894

119891119903( 119894997888rarr

119903) 119871119894( 119894) cos 120579

119894d120596119894

(1 2) = (

1997888rarr 2) =

2minus 1

10038161003816100381610038162minus 1

1003816100381610038161003816

(3)







Image plane


Lens

Image plane

Light source





119868119901= 119896119886119894119886+ sum

119898isinlights(119896119889(119898sdot ) 119894119898119889

+ 119896119904(119898sdot )

120572

119894119898119904

)

(4)



119889(119898sdot )119894119898119889


119904(119898sdot)120572

119894119898119904










119868119901= sum

119898isinlights(119903119888(119898sdot ) 119894119898119889

) (5)








119889mdashonly (

119898sdot ) This


















x

y

z


50 100 150 200 250

50

100

150

200

250

04

045

05

055

06

065

07

075

08

085

09










6 Applications




50 100 150 200 250

50

100

150

200

250

04

045

05

055

06

065

07

075

08

085

09

(a)

50 100 150 200 250

50

100

150

200

250

01

012

014

016

018

02

022

(b)



sum

119898isinlights119903119888119898

= sum

119898isinlights

119868119901119898

(119898sdot ) 119894119898119889

119903119888119898

=

119868119901119898

119908119898(119898sdot ) 119894119898119889

(6)




119898= 1)



50 100 150 200 250

50

100

150

200

250

018

02

022

024

026

028

03

032

034

(a)

50 100 150 200 250

50

100

150

200

250

01

012

014

016

018

02

022

024

026

(b)


Light source


(a)

50 100 150 200 250

50

100

150

200

250

0

005

01

015

02

025

03

035

04

(b)







32 37 42 47 52 570219

02195

022

02205

0221

02215

0222

(a)

32 37 42 47 52 5701605

0161

01615

0162

(b)

32 37 42 47 52 570209

02095

021

02105

0211

02115

(c)

32 37 42 47 52 5701745

0175

01755

0176

01765

(d)


200 300 400 500 600 700 800075085095

Diffusion

(a)

200 300 400 500 600 700 800005

015

025

Spectrum of 705

(b)

200 300 400 500 600 700 800005015025

Spectrumdiffusion

(c)



20 40 60 80 100

200

400

600

800

1000

1200

1400

005

01

015

02

025

03

035

(a)

200

400

600

800

1000

1200

1400

20 40 60 80 100

065

07

075

08

085

09

095

(b)

200

400

600

800

1000

1200

1400

20 40 60 80 100

01

015

02

025

03

035

04

(c)


VoxelSample points

Ray

Viewing point







(a) (b)


(a) (b)










Acknowledgments


References







































Volume 2014

Mining


Journal of



Geophysics







Journal of




Advances in











Geology Advances in





Image plane


Lens

Image plane

Light source





119868119901= 119896119886119894119886+ sum

119898isinlights(119896119889(119898sdot ) 119894119898119889

+ 119896119904(119898sdot )

120572

119894119898119904

)

(4)



119889(119898sdot )119894119898119889


119904(119898sdot)120572

119894119898119904










119868119901= sum

119898isinlights(119903119888(119898sdot ) 119894119898119889

) (5)








119889mdashonly (

119898sdot ) This


















x

y

z


50 100 150 200 250

50

100

150

200

250

04

045

05

055

06

065

07

075

08

085

09










6 Applications




50 100 150 200 250

50

100

150

200

250

04

045

05

055

06

065

07

075

08

085

09

(a)

50 100 150 200 250

50

100

150

200

250

01

012

014

016

018

02

022

(b)



sum

119898isinlights119903119888119898

= sum

119898isinlights

119868119901119898

(119898sdot ) 119894119898119889

119903119888119898

=

119868119901119898

119908119898(119898sdot ) 119894119898119889

(6)




119898= 1)



50 100 150 200 250

50

100

150

200

250

018

02

022

024

026

028

03

032

034

(a)

50 100 150 200 250

50

100

150

200

250

01

012

014

016

018

02

022

024

026

(b)


Light source


(a)

50 100 150 200 250

50

100

150

200

250

0

005

01

015

02

025

03

035

04

(b)







32 37 42 47 52 570219

02195

022

02205

0221

02215

0222

(a)

32 37 42 47 52 5701605

0161

01615

0162

(b)

32 37 42 47 52 570209

02095

021

02105

0211

02115

(c)

32 37 42 47 52 5701745

0175

01755

0176

01765

(d)


200 300 400 500 600 700 800075085095

Diffusion

(a)

200 300 400 500 600 700 800005

015

025

Spectrum of 705

(b)

200 300 400 500 600 700 800005015025

Spectrumdiffusion

(c)



20 40 60 80 100

200

400

600

800

1000

1200

1400

005

01

015

02

025

03

035

(a)

200

400

600

800

1000

1200

1400

20 40 60 80 100

065

07

075

08

085

09

095

(b)

200

400

600

800

1000

1200

1400

20 40 60 80 100

01

015

02

025

03

035

04

(c)


VoxelSample points

Ray

Viewing point







(a) (b)


(a) (b)










Acknowledgments


References







































Volume 2014

Mining


Journal of



Geophysics







Journal of




Advances in











Geology Advances in




119889mdashonly (

119898sdot ) This


















x

y

z


50 100 150 200 250

50

100

150

200

250

04

045

05

055

06

065

07

075

08

085

09










6 Applications




50 100 150 200 250

50

100

150

200

250

04

045

05

055

06

065

07

075

08

085

09

(a)

50 100 150 200 250

50

100

150

200

250

01

012

014

016

018

02

022

(b)



sum

119898isinlights119903119888119898

= sum

119898isinlights

119868119901119898

(119898sdot ) 119894119898119889

119903119888119898

=

119868119901119898

119908119898(119898sdot ) 119894119898119889

(6)




119898= 1)



50 100 150 200 250

50

100

150

200

250

018

02

022

024

026

028

03

032

034

(a)

50 100 150 200 250

50

100

150

200

250

01

012

014

016

018

02

022

024

026

(b)


Light source


(a)

50 100 150 200 250

50

100

150

200

250

0

005

01

015

02

025

03

035

04

(b)







32 37 42 47 52 570219

02195

022

02205

0221

02215

0222

(a)

32 37 42 47 52 5701605

0161

01615

0162

(b)

32 37 42 47 52 570209

02095

021

02105

0211

02115

(c)

32 37 42 47 52 5701745

0175

01755

0176

01765

(d)


200 300 400 500 600 700 800075085095

Diffusion

(a)

200 300 400 500 600 700 800005

015

025

Spectrum of 705

(b)

200 300 400 500 600 700 800005015025

Spectrumdiffusion

(c)



20 40 60 80 100

200

400

600

800

1000

1200

1400

005

01

015

02

025

03

035

(a)

200

400

600

800

1000

1200

1400

20 40 60 80 100

065

07

075

08

085

09

095

(b)

200

400

600

800

1000

1200

1400

20 40 60 80 100

01

015

02

025

03

035

04

(c)


VoxelSample points

Ray

Viewing point







(a) (b)


(a) (b)










Acknowledgments


References







































Volume 2014

Mining


Journal of



Geophysics







Journal of




Advances in











Geology Advances in


x

y

z


50 100 150 200 250

50

100

150

200

250

04

045

05

055

06

065

07

075

08

085

09










6 Applications




50 100 150 200 250

50

100

150

200

250

04

045

05

055

06

065

07

075

08

085

09

(a)

50 100 150 200 250

50

100

150

200

250

01

012

014

016

018

02

022

(b)



sum

119898isinlights119903119888119898

= sum

119898isinlights

119868119901119898

(119898sdot ) 119894119898119889

119903119888119898

=

119868119901119898

119908119898(119898sdot ) 119894119898119889

(6)




119898= 1)



50 100 150 200 250

50

100

150

200

250

018

02

022

024

026

028

03

032

034

(a)

50 100 150 200 250

50

100

150

200

250

01

012

014

016

018

02

022

024

026

(b)


Light source


(a)

50 100 150 200 250

50

100

150

200

250

0

005

01

015

02

025

03

035

04

(b)







32 37 42 47 52 570219

02195

022

02205

0221

02215

0222

(a)

32 37 42 47 52 5701605

0161

01615

0162

(b)

32 37 42 47 52 570209

02095

021

02105

0211

02115

(c)

32 37 42 47 52 5701745

0175

01755

0176

01765

(d)


200 300 400 500 600 700 800075085095

Diffusion

(a)

200 300 400 500 600 700 800005

015

025

Spectrum of 705

(b)

200 300 400 500 600 700 800005015025

Spectrumdiffusion

(c)



20 40 60 80 100

200

400

600

800

1000

1200

1400

005

01

015

02

025

03

035

(a)

200

400

600

800

1000

1200

1400

20 40 60 80 100

065

07

075

08

085

09

095

(b)

200

400

600

800

1000

1200

1400

20 40 60 80 100

01

015

02

025

03

035

04

(c)


VoxelSample points

Ray

Viewing point







(a) (b)


(a) (b)










Acknowledgments


References







































Volume 2014

Mining


Journal of



Geophysics







Journal of




Advances in











Geology Advances in


50 100 150 200 250

50

100

150

200

250

04

045

05

055

06

065

07

075

08

085

09

(a)

50 100 150 200 250

50

100

150

200

250

01

012

014

016

018

02

022

(b)



sum

119898isinlights119903119888119898

= sum

119898isinlights

119868119901119898

(119898sdot ) 119894119898119889

119903119888119898

=

119868119901119898

119908119898(119898sdot ) 119894119898119889

(6)




119898= 1)



50 100 150 200 250

50

100

150

200

250

018

02

022

024

026

028

03

032

034

(a)

50 100 150 200 250

50

100

150

200

250

01

012

014

016

018

02

022

024

026

(b)


Light source


(a)

50 100 150 200 250

50

100

150

200

250

0

005

01

015

02

025

03

035

04

(b)







32 37 42 47 52 570219

02195

022

02205

0221

02215

0222

(a)

32 37 42 47 52 5701605

0161

01615

0162

(b)

32 37 42 47 52 570209

02095

021

02105

0211

02115

(c)

32 37 42 47 52 5701745

0175

01755

0176

01765

(d)


200 300 400 500 600 700 800075085095

Diffusion

(a)

200 300 400 500 600 700 800005

015

025

Spectrum of 705

(b)

200 300 400 500 600 700 800005015025

Spectrumdiffusion

(c)



20 40 60 80 100

200

400

600

800

1000

1200

1400

005

01

015

02

025

03

035

(a)

200

400

600

800

1000

1200

1400

20 40 60 80 100

065

07

075

08

085

09

095

(b)

200

400

600

800

1000

1200

1400

20 40 60 80 100

01

015

02

025

03

035

04

(c)


VoxelSample points

Ray

Viewing point







(a) (b)


(a) (b)










Acknowledgments


References







































Volume 2014

Mining


Journal of



Geophysics







Journal of




Advances in











Geology Advances in


50 100 150 200 250

50

100

150

200

250

018

02

022

024

026

028

03

032

034

(a)

50 100 150 200 250

50

100

150

200

250

01

012

014

016

018

02

022

024

026

(b)


Light source


(a)

50 100 150 200 250

50

100

150

200

250

0

005

01

015

02

025

03

035

04

(b)







32 37 42 47 52 570219

02195

022

02205

0221

02215

0222

(a)

32 37 42 47 52 5701605

0161

01615

0162

(b)

32 37 42 47 52 570209

02095

021

02105

0211

02115

(c)

32 37 42 47 52 5701745

0175

01755

0176

01765

(d)


200 300 400 500 600 700 800075085095

Diffusion

(a)

200 300 400 500 600 700 800005

015

025

Spectrum of 705

(b)

200 300 400 500 600 700 800005015025

Spectrumdiffusion

(c)



20 40 60 80 100

200

400

600

800

1000

1200

1400

005

01

015

02

025

03

035

(a)

200

400

600

800

1000

1200

1400

20 40 60 80 100

065

07

075

08

085

09

095

(b)

200

400

600

800

1000

1200

1400

20 40 60 80 100

01

015

02

025

03

035

04

(c)


VoxelSample points

Ray

Viewing point







(a) (b)


(a) (b)










Acknowledgments


References







































Volume 2014

Mining


Journal of



Geophysics







Journal of




Advances in











Geology Advances in


32 37 42 47 52 570219

02195

022

02205

0221

02215

0222

(a)

32 37 42 47 52 5701605

0161

01615

0162

(b)

32 37 42 47 52 570209

02095

021

02105

0211

02115

(c)

32 37 42 47 52 5701745

0175

01755

0176

01765

(d)


200 300 400 500 600 700 800075085095

Diffusion

(a)

200 300 400 500 600 700 800005

015

025

Spectrum of 705

(b)

200 300 400 500 600 700 800005015025

Spectrumdiffusion

(c)



20 40 60 80 100

200

400

600

800

1000

1200

1400

005

01

015

02

025

03

035

(a)

200

400

600

800

1000

1200

1400

20 40 60 80 100

065

07

075

08

085

09

095

(b)

200

400

600

800

1000

1200

1400

20 40 60 80 100

01

015

02

025

03

035

04

(c)


VoxelSample points

Ray

Viewing point







(a) (b)


(a) (b)










Acknowledgments


References







































Volume 2014

Mining


Journal of



Geophysics







Journal of




Advances in











Geology Advances in


20 40 60 80 100

200

400

600

800

1000

1200

1400

005

01

015

02

025

03

035

(a)

200

400

600

800

1000

1200

1400

20 40 60 80 100

065

07

075

08

085

09

095

(b)

200

400

600

800

1000

1200

1400

20 40 60 80 100

01

015

02

025

03

035

04

(c)


VoxelSample points

Ray

Viewing point







(a) (b)


(a) (b)










Acknowledgments


References







































Volume 2014

Mining


Journal of



Geophysics







Journal of




Advances in











Geology Advances in


(a) (b)


(a) (b)










Acknowledgments


References







































Volume 2014

Mining


Journal of



Geophysics







Journal of




Advances in











Geology Advances in





Acknowledgments


References







































Volume 2014

Mining


Journal of



Geophysics







Journal of




Advances in











Geology Advances in










Volume 2014

Mining


Journal of



Geophysics







Journal of




Advances in











Geology Advances in

Research Article Modeling the Reflectance of the Lunar...

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Transcript of Research Article Modeling the Reflectance of the Lunar...